Description

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1906 edition. Excerpt: ...enter. Everyone will naturally prefer the methods with which he is most familiar; but I think that it may be safely afirmed that in the majority of cases in this field the advantage derived from the use of the quatemion is either doubtful or very trifling. There remains a residuum of cases in which a substantial advantage is gained by the use of the quaternionic method. Such cases, however, so far as my own observation and experience extend, are very exceptional. If a more extended and careful inquiry should show that they are ten times as numerous as I have found them, they would still be exceptional. We have now to inquire what we find in the Ausdehnungslehre in the way of a geometrical algebra, that is wanting in quaternions. In addition to an algebra of vectors, the Ausclehnu/ngslehre affords a system of geometrical algebra in which the point is the fundamental element, and which for convenience I shall call Grassma.nn's algebra of points. In this algebra we have first the addition of points, or quantities located at points, which may be explained as follows. The in which the capitals denote points, and the small letters scalars (or ordinary algebraic quantities), signifies that and also that the centre of gravity of the weights a, b, 0, etc., at the points A, B, C, etc., is the same as that of the weights e, f, etc., at the points E, F, etc. (It will be understood that negative weights are allowed as well as positive.) The equation is thus equivalent to four equations of ordinary algebra. In this Grassmann was anticipated by Mobius (Barycentrtscher Catcul, 1827). - We have next the addition of finite straight lines, or quantities located in straight lines (Liniengrosscn). The meaning of the will perhaps be understood most readily, ...show more